Conformally invariant scaling limits: an overview and a collection of problems

By Oded Schramm. 2006

148 citations recorded by the source corpus.

Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.

Subfields: Condensed Matter Physics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9adfc70b7def8151cbf9a01df5e8d50e97e1c1ef

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In Condensed Matter Physics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Towards conformal invariance of 2D lattice models (2007, background): “Finally, there are many other open questions related to conformal invariance, some of which are discussed in Oded Schramm’s paper [34] in these proceedings.”
Random geometry on the sphere (2014, background): “A key result [31], which solves a conjecture of Oded Schramm [44], states that”
Geodesics in large planar maps and in the Brownian map (2008, background): “The problem of establishing a convergence of the type (1) was raised by Schramm [36] in the setting of triangulations.”
Uniqueness and universality of the Brownian map (2011, background): “This solves an open problem that has been stated first by Oded Schramm [26] in the particular case of triangulations.”
Scaling limits of random trees and planar maps (2011, methodology): “These ideas appeared in the pioneering paper of Chassaing and Schaeffer [12] and in the subsequent work of Markert andMokkadem [37] in the case of quadrangulations, and a little later in Schramm [48], who gave a precise form to the question of the existence of a scaling limit for large random triang”
Schramm-Loewner Evolution (2007, background): “[72] O. Schramm (2007), Conformally invariant scaling limits: an overview and a collection of problems, Proc.”
Imaginary geometry II: Reversibility of SLEκ(ρ1;ρ2) for κ∈(0,4). (2016, unknown): “In particular, we will give a new proof of the time-reversal symmetry of ordinary SLEκ for κ < 4 (inspired (1)This was the final problem in a series presented by Schramm at ICM 2006 [Sch11].”
Reversibility of chordal SLE (2008, background): “Scott Sheffield proposed that the reversibility can be derived from the relationship with the Gaussian free field [10].”